For as long as biologists have been computing alignments of sequences, the question of what values to use for scoring substitutions and gaps has persisted. In practice, substitution scores are usually chosen by convention, and gap penalties are often found by trial and error. In contrast, a rigorous way to determine parameter values that are appropriate for aligning biological sequences is by solving the problem of Inverse Parametric Sequence Alignment. Given examples of biologically correct reference alignments, this is the problem of finding parameter values that make the examples score as close as possible to optimal alignments of their sequences. The reference alignments that are currently available contain regions where the alignment is not specified, which leads to a version of the problem with partial examples.

In this dissertation, we develop a new polynomial-time algorithm for Inverse Parametric Sequence Alignment that is simple to implement, fast in practice, and can learn hundreds of parameters simultaneously from hundreds of examples. Computational results with partial examples show that best possible values for all 212 parameters of the standard alignment scoring model for protein sequences can be computed from 200 examples in 4 hours of computation on a standard desktop machine. We also consider a new scoring model with a small number of additional parameters that incorporates predicted secondary structure for the protein sequences. By learning parameter values for this new secondary-structure-based model, we can improve on the alignment accuracy of the standard model by as much as 15% for sequences with less than 25% identity.